In the LHC each proton reaches an energy of 7 TeV. Firstly, we introduce some basic calculations in Special Relativity and afterwards we present a more detailed calculation about energy of collisions in relativistic terms.

Let´s take a look at the relativistic parameter γ (gamma) when the proton reaches that energy.

In this case, the expression for Kinetic Energy is:

Ek =γ·m0·c2 - m0·c2

or

Ek = m0·c2(γ-1) (1)

The mass of proton is 938,3 MeV/c2

So,m0·c2 = 9,383·10-4 TeV

So (1): 7 TeV = 9,383·10-4(γ-1)

γ ~ 7460

γ>>1 , therefore we are nearing Special Relativity.

We can now verify the of the proton with that energy comes close to that of the speed of light.

In effect, γ = 1/[1- (v/c)2]1/2

with γ = 7460⇒v = 0,999999991·c , so:v ~ c

The use of energy units for other magnitudes is normal in the Physics of High Energies.

Therefore the energy at rest of the proton will be:

E0 =m0·c2⇒ E0 = 1,67·10-27kg·(3·108m/s)2⇒ E0 = 1,503·10-10 J

Since, m0 = E0/c2 , the rest mass for the proton is: m0= 938,3 MeV/c2

That mass can also be expressed:m0=0,0009383 TeV/c2

Even though it´s a debatable concept that it will not be argued here, the energy reached by the proton in the LHC could be expressed as “relativistic mass”:

m = E/c2⇒ m = 7 TeV/c2

Compare this value with m0=0,0009383 TeV/c2 to see the increase experimented.

We could also get the value of the momentum of the proton in relative terms:

E2 = (p·c)2 + E02⇒ p·c = (E2 - E02)1/2

p·c = (72 - 0,00093832)1/2⇒ p ~ 7 TeV/c

The value of the momentum of the proton is another important parameter in the accelerator.

Xabier CidVidal, PhD in experimental Particle Physics for Santiago University (USC). Research Fellow in experimental Particle Physics at CERN from January 2013 to Decembre 2015. Currently, he is in USC Particle Physics Department ("Ramon y Cajal", Spanish Postdoctoral Senior Grants).

Ramon Cid Manzano, secondary school Physics Teacher at IES de SAR (Santiago - Spain), and part-time Lecturer (Profesor Asociado) in Faculty of Education at the University of Santiago (Spain). He has a Degree in Physics and in Chemistry, and is PhD for Santiago University (USC).